A datum reference frame (DRF) is a coordinate system used to locate and orient part features. Constructing a DRF from a set of datum features is a complicated process involving: a) specifying a valid combination and the precedence of the datum features which define the DRF; b) developing datum from datum features of the part; and c) determining the position and orientation of the DRF from the datums. We develop a geometric theory for establishing DRFs. The theory is based on the observation that a datum feature such as a plane, a cylinder or a sphere has a symmetry subgroup G/sub 0/ under the action by the group SE(3) of rigid motions in R/sup 3/. Thus, the configuration space of a datum feature can be identified with the homogeneous space SE(3)/G/sub 0/, and the problem of datum development can be posed as a minimization problem in SE(J)/G/sub 0/. We give conditions under which a datum feature qualifies to be a secondary or a tertiary datum. We present a sequential procedure that transforms the primary, secondary and tertiary datum problems as a minimization or a constrained minimization problem in the homogeneous spaces of SE(3). We develop simple algorithms to solve these problems, and give simulation results illustrating efficiency and simplicity of the approach.

We first propose a hybrid localization/inspection/machinability problem. Next, we formulate the hybrid problem using differential geometric theory and the minimax method. Then, we develop a methodology for treating localization, online inspection and machinability of workpieces simultaneously. Using the geometric properties of the hybrid problem, the hybrid problem is decoupled into a (symmetric) localization/inspection problem and a machinability problem. Then both problems are formulated as constrained optimization problems and are solved by a sequence of linear programming problems. Finally, we present simulation results to demonstrate the efficiency of our method for the hybrid problem.

A datum reference frame is a coordinate system used to locate and orient port features. Constructing a datum reference frame from a set of features is a complicated process involving: 1) specifying a valid combination and the precedence of the datum features which define the datum reference frame; 2) developing from datum features of the part; and 3) determining the position and orientation of the datum reference frame from the data. In this paper, we develop a geometric theory for establishing datum reference frames, and present a sequential procedure that transforms the primary, secondary and tertiary datum problems as a minimization or a constrained minimization problem. We develop simple algorithms to solve these problems, and give simulation results illustrating efficiency and simplicity of the approach.

We first present a unified geometric formulation of form tolerances using an extended least-squares (E-LSQ) approach. Then, using properties of configuration space of symmetric features an iterative algorithm is developed to solve the E-LSQ problem. Simulation results show that the algorithm not only keeps computational efficiency of the least-squares approach, but also possesses of computational accuracy of min-max algorithms. The outstanding properties of the geometric formulation include: 1) coordinate-free representation; 2) in conformance to the tolerance standard; and 3) easy implementation.

This paper defines a hybrid localization/envelopment problem, develops a formulation of the hybrid localization/envelopment problem, and presents a simple algorithm for computing its solutions. First, we show that when the finished surfaces of a workpiece are inadequate to fully constrain the rigid motions of the workpiece, then the remaining set of free motions must form a subgroup G/sub 0/ of the Euclidean group SE(3). This allows us to decompose the hybrid problem into a (symmetric) localization problem on the homogeneous space SE(3)/G/sub 0/ and an envelopment problem on G/sub 0/. The geometric properties of SE(S)/G/sub 0/ are used to convert the envelopment problem into a nonlinear programming problem with a convex objective function, which is then solved using techniques from nonlinear programming. Finally, we present simulation results to illustrate the effectiveness of our method for the hybrid problem.

Workpiece localization plays a vital role in automation of many important manufacturing processes, such as workpiece setup, refixturing and dimensional inspection. In this paper, we provide a unified treatment of three geometric algorithms for workpiece localization, and develop new techniques to make these local algorithms globally convergent. We also study and analyze, along with extensive simulation results, their performance with respect to convergence and computational efficiency. Finally, we present a method for analyzing reliability of localization solutions and give a lower bound on the number of measurement points needed for reliable recovering of Euclidean transformations.

The rapid proliferation of coordinate measuring machines (CMM) triggered the need for precise and rigorous formulations of each tolerance concept. In this paper, we employ the concept of configuration space of symmetric features to define each type of datum features and orientation tolerances. The establishment of datum features and orientation tolerances are formulated in an linear programming approach. Using properties of Lie groups and Lie algebra, a simple, unified and coordinate free algorithm for datum establishment and orientation tolerance is developed. The results of the algorithm comply to the definitions stipulated in the standard ANSI Y14.5M.

Accurate and efficient localization of symmetric features plays an important role in dimensional inspection of machined parts and machining of partially finished workpieces. We present a geometric theory for efficient and accurate localization of symmetric features. First, we show that the configuration space of a symmetric feature can be naturally identified with the homogeneous space SE/sub (3/)/G/sub o/ of the Euclidean group SE/sub (3/), where G/sub o/ is the symmetry group of the feature. Then, we explore the geometric structure of the homogeneous space and present a simple and unifying algorithm for symmetric localization. Finally, we give simulation results illustrating several unique features of the algorithm: 1) implementational simplicity; 2) robustness with respect to initial conditions; 3) high accuracy in computed results; and 4) computational efficiency.

We present a unified geometric theory for localization of three types of workpieces: 1) general three-dimensional (3D) workpieces where points from the finished surfaces fully constrain the rigid motions of the workpieces; 2) symmetric workpieces; 3) partially machined workpieces where points from the finished surfaces are inadequate to fully constrain the rigid motions of the workpieces. Applications of the study include workpiece setup, refixturing and dimensional inspections in a flexible manufacturing environment. First, we formulate the localization problem for a general 3D workpiece and study the mathematical properties of the underlying problem. We discuss an iterative approach for solving the general localization problem and show how different considerations in updating the Euclidean transformation lead to various geometric algorithms. Then, we extend the localization techniques to symmetric workpieces and partially machined workpieces and present a simple algorithm for each of the problems. Finally, we present simulation results showing convergence and robustness properties of the various geometric algorithms.

Workpiece localization, with novel applications such as workpiece setup, refixturing and dimensional inspections, is a problem of permanent importance in manufacturing. Using the popular least square formulation, several geometric algorithms have been developed for workpiece localization over the last few years. In this paper, we analyze and compare the performance of three localization algorithms based on the following criteria: (a) robustness with respect to variations in initial conditions; (b) accuracy of computed results; and (c) computational efficiency. We develop an approach for improving the robustness of the algorithms for workpieces with sculptured surfaces for which the region of convergence is typically small. Based on simulation results, we also discuss sensitivity of the algorithms with respect to the number of measurement points and give a lower bound on this number for recovering a Euclidean transformation with certain accuracy.

In this paper an efficient method for automatic probing and localisation of 3D workpiece arbitrarily fixed to the machine table of a five-axis milling machine is presented. First, by formulating localisation problem as minimizing a least-square question an explicit solution is obtained. Based upon this solution and sensitivity analysis, we present some guidelines and develop a CAD-based automatic probing strategy. By means of the knowledge about the approximate location of workpiece computed by 3D localisation algorithm as estimator, together with the CAD model of workpiece, we then generate online a new collision free probing trajectory to probe additional point on a neighbourhood of the previous sampling point. Utilizing the newly measurement point, we improve the Euclidean transformation which in turn is used together With the CAD model, to generate another collision free probing path to probe more points on the surfaces of workpiece. This process continuous until the workpiece is accurately located. Experimental results show that this algorithm overcomes the need of home surface identification process and is thus suitable for real-time implementation in manufacturing or inspection process.